Global smooth fibrations of ℝ3 by oriented lines

Abstract

A smooth fibration of ℝ 3 by oriented lines is given by a smooth unit vector field V on ℝ 3 all of whose integral curves are straight lines. Such a fibration is said to be nondegenerate if dV vanishes only in the direction of V. Let ℒ be the space of oriented lines of ℝ 3 endowed with its canonical pseudo-Riemannian neutral metric. We characterize the nondegenerate smooth fibrations of ℝ 3 by oriented lines as the closed (in the relative topology) definite connected surfaces in ℒ. In particular, local conditions on ℒ imply the existence of a global fibration. Besides, for any such fibration the base space is diffeomorphic to the open disc and the directions of the fibers form an open convex set of the two-sphere. We characterize as well, in a similar way, the smooth (possibly degenerate) fibrations.